Accelerated Stabilization of Switched Linear MIMO Systems using Generalized Homogeneity

This paper addresses the problem of exponential and accelerated finite-time, as well as nearly fixed-time, stabilization of switched linear MIMO systems. The proposed approach relies on a generalized homogenization framework for switched linear systems and employs implicit Lyapunov functions for control design, covering both common and multiple Lyapunov function settings. Linear matrix equations and inequalities are derived to characterize the dilation generator and to synthesize the controller gains. Robustness of the resulting control laws with respect to system uncertainties and external disturbances is analyzed. The effectiveness of the proposed approach is illustrated through numerical examples.

💡 Research Summary

This paper tackles the problem of stabilizing switched linear multiple‑input multiple‑output (MIMO) systems with performance requirements ranging from exponential convergence to accelerated finite‑time and nearly fixed‑time convergence. The authors introduce a unified design methodology based on a generalized homogeneity framework. By embedding a linear state‑feedback term K₀x together with a homogeneous correction term that depends on a dilation operator d(s)=e^{sG_d}, the closed‑loop dynamics can be rendered d‑homogeneous of a prescribed degree μ. The sign of μ determines the convergence regime: μ<0 yields global finite‑time stability, μ=0 recovers standard exponential stability, and μ>0 provides nearly fixed‑time stability (i.e., a uniform bound on the settling time).

The paper first reviews the necessary background on switched systems, Lyapunov stability notions, and the theory of homogeneous vector fields. It then formalizes the concept of a d‑homogeneous switched system (Definition 5) and proves that if each subsystem is d‑homogeneous of the same degree, the overall switched system inherits this property (Theorem 5). A key algebraic condition (Corollary 1) shows that a linear switched system ˙x=C_{σ(t)}x is d‑homogeneous of degree μ if and only if C_i G_d – G_d C_i = μ C_i for every mode i.

Two parallel controller synthesis routes are developed. The first uses a common implicit Lyapunov function V(x)=xᵀPx. By enforcing the differential inequality \dot V ≤ –κ V^{α} with α related to μ, the authors obtain explicit formulas for K₀ and the homogeneous gain K_d. The second route employs multiple Lyapunov functions V_i(x)=xᵀP_i x, one per mode, together with an average/minimum dwell‑time condition that guarantees uniform decrease of the Lyapunov values across switches. Theorems 8 and 9 provide the sufficient LMI conditions for this multi‑Lyapunov approach.

The synthesis of the feedback gains hinges on solving a special Sylvester‑type matrix equation X A – A X = X. Lemma 1 shows that any solution X is nilpotent, and Lemma 2 guarantees existence of a solution for any prescribed nilpotent X. This property enables the construction of the generator matrix G_d and the feedback matrices via linear matrix equations and LMIs, offering a systematic and numerically tractable design procedure.

Robustness is addressed by incorporating an external disturbance term E_i ω(t,x) into the plant model. The authors derive a perturbed Lyapunov inequality \dot V ≤ –κ V^{α} + γ‖ω‖ and provide explicit bounds on γ that preserve the desired convergence rate despite uncertainties. This analysis demonstrates that the proposed controllers are robust to both matched and unmatched disturbances.

Numerical simulations illustrate the theory on a three‑mode switched linear system. Three cases are examined: μ=–0.5 (accelerated finite‑time), μ=0 (exponential), and μ=0.8 (nearly fixed‑time). The state trajectories confirm the predicted convergence behavior, and the addition of bounded disturbances shows that the performance degradation remains within the analytically derived robustness margins.

In summary, the paper contributes a comprehensive framework that unifies exponential, finite‑time, and nearly fixed‑time stabilization of switched linear MIMO systems through generalized homogeneity. It provides both common‑Lyapunov and multiple‑Lyapunov constructions, explicit LMI‑based gain synthesis, and robustness guarantees, thereby extending the toolbox for control of hybrid systems with stringent transient specifications. Future directions suggested include extensions to nonlinear switched dynamics, time‑delay systems, and data‑driven adaptive schemes.